MASSACHUSETTS INSTITUTE OF TECHNOLOGY LINCOLN LABORATORY

THE EFFECTS OF INTERFERENCE ON MONOPULSE PERFORMANCE

R. J. McAULAY

TECHNICAL NOTE 1973-30

1 AUG 1973

LEXINGTON MASSACHUSETTS

.

CONTENTS

List of Illustrations

1. Introduction

2. Interference Effects on Estimate Bias

3. Special Case: Real Antenna Patterns

4. Interference Effects on Estimate Variance

5. Conclusions

References

Appendix

Acknowledgment

. .

~.,

,..111

iv

1

2

8

21..

27

29

30

31

LIST OF ILLUSTRATIONS

Figure

l(a) EQUIVALENT AZIMUTH USINGTHE MONOPUI.,SE FUNCTION. . . . . . . . . . . . . . . . . . . . . . . ...12

l(b) PROBABILITY Y DISTRIBUTIONFUNCTION OF EQUIVALENT AZIMUTH. . . . . . . . . . . . . . ...12

2. AZIMUTH ESTIMATE IN THE PRESENCE

OF INTERFERENCE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3. AZIMUTH ESTIMATE IN THE PRESENCE

OF INTERFERENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4. VARIANCE OF THE AZIMUTH ESTIMATE

IN THE PRESENCE OF INTERFERENCE . . . . . . . . . . . . . . . . 26

iv

,,

1. INTRODUCTION

There are several competing factors that limit the performance of the

monopulse azimuth estimation accuracy in a noise and interference background.

The first is due to the fading phenomenon that results in a degradation in the

signal-to -noise ratio (SNR) as the target and interference signals approach

the out of phase condition. Pruslin [1], in a simulation of the receiver noise

effects on monopulse performance, has observed that the estimator becomes

biased for low values of the SNR. Browne [2], in an interference-free analysis,

has calculated this bias exactly. for all values of the SNR.

Closely related to the bias introduced by fading, is what we shall refer

to as the azimuthal bias that results when the interferer and the target signal

sources are located at different azimuths. Finally, the presence of inter -

fering signals causes the variance of the estimate to increase.

In this note we shall study mainbeam and sidelobe interference and

noise effects simultaneously and use an analysis similar to Browne’s to

compute an exact expression for the estimator bias. We specialize our results

to real antenna patterns and obtain an expression for the bias that holds for

the high and low SNR situations, hence, we obtain the azimuthal bias and the

fading bias simultaneously. Numerical results are given for some typical

operating conditions and it is shown that the bias can be a large fraction Of

a beamwidth.

Some further analytical results are obtained for the case of sidelobe

interference for real antenna patterns. In this case too, it is possible for an

ATCRBS (Air Traffic Control Radar Beacon System) interferer that is located

in a large near-in sidelobe to cause bias errors that are a large fraction of

a beamwidth.

1

In addition to introducing a deterministic bias to the azimuth estimate,

interference causes the random error to have an increased variance.

We obtain a general expression for the variance that applies to the mainbeam

and sidelobe interference cases using an analysis similar to that developed

by Sharenson [3] who analyzed the noise-only case. Numerical results arc

given that show that the variance also depends critically on the relative

phase between the target and interference signals.

These general expressions for the bias and the variance of the mono-

pulse estimate provide the tools needed to thoroughly understand the effects

of interference. These are essential in the evaluation of signal processing

techniques. to overcome the effects of interference. The leading candidate

at the moment, is the monopulse interference detector and data editing

s theme followed by an outlier te st. Although this is the subject of a sepa-

rate paper [4] the results of the present study are the background for the

succeeding analysis.

2. INTERFERENCE EFFECTS ON ESTIMATE BIAS

We restrict our attention to sum-difference (even-odd) monopulse pro-

cessing that is performed on a sampled-data basis. Assuming mixer pre-

amplifiers at the output of the sum and difference beams, the received si,g -

nals in the presence of interference are modelled by

‘.,

jvB j VIyi = A=e Gi(e~) + Ale Gi(E!l) + n. i=l,2

1(1)

where yl, y2 refer to the outputs of the sum and difference beams; As, q~,

es and AI, t+iI, f)l are the amplitude, phase, and azimuth of the target and interference

respectively; Gl( ), G2( ) are the antenna patterns of the sum and difference beams,

and these may be complex in general; nl, n2 are the independent, zero mean Gaussian

noise samples due to the mixer preamps whose real and imaginary parts have vari -

2ante o

An azimuth estimator that is often used in practice is given by

the relation

E(j) = Re(y2/yl) (2)

where the monopulse function, E ( e), is simply the normalized difference

pattern,

E(e) = G2(9)/Gl(9) (3)

In most cases of practical interest, this function is well approximated by

a linear characteristic, hence we can write

E(13) = k(6/BB) (4)

where 9B

refers to the 3 dB beamwidth of the antenna patterns and k is a

standard parameter that arises in the characterization of monopulse systems.

Usually 1 S k S 2 with k . 1.5 being a typical value. When this approximation

is valid, the azimuth estimate is readily generated as

(5)

We will incorporate (3B into our definition of e and hence express all our

results in units of a 3 dB beamwidth.

Our goal is to compute the mean and variance of (5) when inter-

ference is present in the sum and difference signals as given by (l). We

begin by letting

j US j VI .Di

Ui = A~Gi(9~)e + AIGi(fI1)C = Vi# (6)

Since ni are complex Gaussian variates, they can be expressed as

ni . Ni expj a; where Ni are Rayleigh and O; are uniform random variables.

Using these definitions in (1) we have

(7)

4

Since ~i and W> are independent and uniform, then so too are o ;-pl~ ~,,

We apply these relations to evaluate (5) by writing

Re (L\. ‘e(Y~Y~)

.

\yl) IYJZ

1( )[ 1V2COS(B2 - 81) + NZc05~2 VI + NlCOSCI1 + ‘~sin(~2- Bl)+N2sin% ‘lsin~

.L L

‘1 + 2vlNlc0s% + ‘1 (8)

Following Brov.me [2] we first average (8) over IY2. Since this is uniform on

(O, 2n), we have

E (k$) =V2COS(~ - 81) (Vl + NICOE~) + V2N1sin(~2 - ~l)sintrl

%

(9)

V; + N: + 2T1N1COSCY1

Next he averages over O’1. In the Appendix this is shown to give

E (k~) =

al ‘2

o if N1>V1

‘2(lo)

—cOs (@2 - PI) if N1<V1

‘1

5

Since N1 is the magnitude of a complex Gaussian random variable having

2.variance 20 , It has the Rayleigh distribution

N;2

()

‘1p(N1) = — exp - ‘2

tiu 40

Then averaging (10) over N1 gives

‘2E(kil) = ~ COS(D2-5 ) A

1 1 fill

v. [

(11)

v~1

()

- N:

N1 exp7

dN

(1 401

(12)

It should be noted that when interference is absent, AI = O

and from (6) we see that ~1 . ~ and (12) reduces to the same expression

obtained by Brovme. However, we are now in a position to determine the

effects of interference on the azimuth bias. To do this we need to evaluate

VI, V2/V1 and F3z - f?l from (6).

Before per formi~ this evaluation wc first note that the target of

interest lies within the mainbeam of the antenna, hence we may assume that

G1(fl~) and G2(6~) are real functions of 9 Whereas the sum and differences“

beams are in phase over the antenna beamwidth, small errors in the ampli-

tude and phase tapers render them complex in the sidelobes. Since the

6

interference can be located in the mainbcam or within the sidelobes of the

antenna, then Gi(61) must be considered to be complex in general. To make

this explicit we write

Gi(~) = Ai ($) expyi (91)

Then substituting this into (6) we can easily show that

1Vi = A:G~ (o.) + 2A~A1Gi (o.) Ai (91) ~OS [v+ Yi(el)]

(13)

(14)

-1 AIAi ($) sinw+ yi (eI)ei = ,0~ + tan

A~Gi(8~) + A#i(8~cos [CO+ Yi($)](15)

where v = q - co. is the relative phase between the tar~et and intcrfercncc

signals. Using these expressions in (12) and performing straightforward

but tedious trigonometric manipulations we can show that the first moment

of the azimuth estimate is

(16)

where p . A1/A~ represents the intcrfcrcn cc-to-signal ratio (ISR) prior

to any processing and

7

where

-1y . tan

Az

2 ‘1l+ZO+COS(Q+Y1)+P y

1‘1

G2(B~) Al sinl’l +G1(Eis) A2 sill y2

G2 (es) ‘1 Cos ‘1+- G1(e~) A2cos y2 1

\

(17)

(18)

These are the key expressions we need to fully understand the deterministic

aspect of the azimuth error introduced by interference.

3. SPECIAL CASE: REAL ANTENNA PATTERNS

The results that have been obtained to this point are free of any

approximating assumptions and apply to the general case of complex antenna

patterns. Unfortunately it is difficult to draw analytical conclusions from

these expressions because of the large number of parameters that would have

to be examined simultaneously. Results for the general case will be obtained

using a computer simulation. There is one case of considerable practical

8

I

I

.

importance that we can explore analytically. This is the situation in which the

antenna patterns are real, which is a reasonably accurate model for the high

near-in sidelobes of the antenna. For this case wc assume that

Ai(91) si,, Ti(131) % O i=l,2 (19)

and then define

Gi (131) = Ai (fII) CCJSYi ( 91) (20)

which we note arc real functions for all values of tll. Using this assumption,

and (17) become

iie.i0

[[

+;(0s) Gl(f+) ~ G;(eI)l-exp- 1 + 2Pmc0scp+ p

2.2 G; (es) IIwhere now

0G1(131) ~ G:(B1)

1+2PG1(9~) co’ ‘“+ pG:(9~)

(21)

(22)

9

We recall that p = A1/A~ represents the interference-to-signal ratio (ISR) prior

to any processing. The measured ISR at the outputs of the antenna ports is

given by

A1G1(91) _ G1(51)P. ‘

A~G1(8~) - p G1 (9~)(23)

Although both definitions of ISR are important parameters of the system, it

is convenient to rewrite (21) and (22) in terms of p . In this case the mean0

value of the azimuth estimate is given by

j= $ 1[ -A:G; (13~)1 - exp

0II

(1 + Zpocosw+ p:) (24)

2 U2

where now

G2(8~)

[

G2(13~)G2(eI)

1

z G2(e1)

—+p ——G1(8~)

cos~ip —o G1(13&) + G1( 61) o G1(el)

kg =0

1 + 2poc0sm+ p:

It is interesting that the performance depends

E(fl) = G2(0/G1(13). In most cases of interest

(25)

only on the monopulse function

the target will be located within

the 3 dB beamwidth of the antenna. Therefore, using (3) and (4) we can write

G2(8~)

Gl(e~) = ‘es(26)

.

10

where 9~ is measured in 3 dB beamwidths. In the case of mainbeam inter-

ference this linearity property will also be valid, but in more general cases

E(81) will simply take on the values of the normalized difference pattern. 10

handle this situation we define an equivalent az.i]muth

Gz(O1)

F1=~—k Gl(t31)

(27)

and note that whenever el is within the 3 cll> beamwidth of the antenna 9 = 6I 1“

In Figure l(a) we have plotted the equivalent azimuth for a monopulse antenna

configuration that has a 4° beamwidth and achieves a -20 d13 peak sidelobe level

on both the “sum” and “difference” beams. Assuming that ~1 is uniformly

distributed in (-TI, n), Figure l(b) shows the probability distribution function

of the equivalent azimuth. This shows that “most of the time” the equivalent

azimuth will be less than 2 beamwidths, hence, for the purpose of analysis,

wc can study the cases of mainbeam and sidelobe interference simultaneously.

Then using (26) and (27) in (25) we can express the first moment of the azimuth

estimate as

xxe=e I1- exp

o

A:G:(fJ~)

(l+2poc0sw+p

202~2) II

x e~+Do(e~+F1)cos(, +p:F1e. =

1+200 CO SW+O:

(28)

(29)

11

20

0

-2C

4° BEAM WIDTH

-20-dB PEAK SIDE LOBES

(o)

1 I I I

--@EEL

I I I

-80 -60 -40 -20 0 20 40 60 80

AZIMUTH (deql

Fig. la. Equivalent azimuth using the monopulse function.

!.0 r 1

J’”

-6 -4 -2 0 2 4 6

EQUIVALENT AZIMUTH ( Beomwid!hs)

Fig. lb. Probability distribution function of equivalent azimuth.

12

It is from this equation that we can begin to make some general

statements regarding the performance of the azimuth estimator. We shall

first show how the results arc consistent with th~se previously obtained and

then go on to consider cases that have not yet been explored in the literature.

(a) Case 1: No Interference

When there is no interference, AI . 0, hence P = O and fromo

(28) and (29) the average value of the estimate is

[ IIA;G:(3~)

l-cxp -

2D2(30)

This is the result obtained by Browne [2] and describes analytically Pruslin’s

observation [1] that as the SNR, A sz G ,2 (9 s)/2 o 2,dccreascs, the monopulsc

estimate is biased towards the antenna bores ight.

We have tabulated (30) in Table 1 from which it can be seen that

for SNR’S above 8 ~B, the bias can be considered to bc negligible. Since in

the Air Traffic Control (ATC) system the SNR will be at least 20 dB, this

effect will be insignificant unless fading occurs, as willbe discussed in the

next section.

Table 1. Variation of Bias with SNR

SNR (dB)

o3

3.6

4.766.6

7.258.39

8.89.64

iie-e

s

eB

0.370.135

0.1

0.05

0.01

0.005

0.001

0.0005

0.0001

(b) Case 2: Fading

If the target and interference signals arrive from the same azimuth

(multipath from a flat earth for example) then El ❑ $ = 8~ and (28) and (29)

reduce to

xe=e

s[

A: G; (85)l-exp- (1 + 2p&0smi P:)

2’02 II

(31 )

This demonstrates the existence of a fading bias that can occur even at large

SNR when the interference

tends to cancel its energy.

fading will be negligible as

This will be the case if

signal occurs out of phase with the target and

From Table 1 wc conclude that the bias due to

long as the effective SNR is greater than 8 d13.

14

For a 20 dB

.752 p. ~

(c) Case 3:

20 ‘

SNR, this shows

1.25.

Azimuthal Bias

+ Zpocosm + & ~ 6.3 (3~,

that fading could become a problem only if

. .Next, we consider the case in which the SNR is large and the

interference is of low enough power that the exponential term in (28) is

negligible. Then (29) reduces to

x es + (es +F1)oocOsm + p. ~‘F

e= (33)

1 + Zpocosw+oz0

This result is sufficiently general to describe the cases of mainbcam and

xsidelobe interference. Notice that when OS = 81 = ~1 then 8 ‘ es and the

estimator is unbiased. T’hercfore a bias is obtained only when f3~ # e ~ ,

namely when the target and interference signals are separated in azimuth.

It is for this reason that wc refer to this effect as the azimuthal bias.

To obtain some physical appreciation for the behaviour of the

estimator wc consider a specific case of interest. We assume the

target is located on boresight at 20 dB SNR. Therefore, es = O and

A~G; (0)/2u2 = 100. Wc next let the interferer be located at the edge of the

3 dB beamwidth so that &l = ff, = .5. TO within a good approximation the sum

beam is quadratic over the bcamwidth, hence G1(9) = 1 - 1.7132. Since the

15

output interference-to-signal ratio is PO = pG1(O1)/G1(e~) where p = A1/A~,

wc see that p = . 707p Equation (33) is plotted as a function of the relative0

phase angle for various values of p. The results are shown in Figure 2.

Equation (28) was also plotted but there was no discernible difference in

resulting curves at least for the values of p we used. In Figure 3 these

the

equations were again plotted for P. nearer to unity and the fading effect can

be observed at the out of phase condition. Since there is only a small range

of values for which this effect is observable and since its effect is, if anything,

beneficial, we shall neglect the fading bias in the rest of our work and restrict

our attention to the azimuthal error as described by (29).

The bias in the estimate can be written as

—P. + Cos m

b(v) : : - El, = (61 - es) P.

1 + 2POCOSO+P20

where PO and 91 are given by (23) and (27) respectively.

(34)

Therefore for small

t. 0x

values of p. , ~hilc for Iargc values B . ~1 which shows how the inter-s’

ferer “captures” the estimate. For intermediate values, curves like those

shown in Figures 2 and 3 are obtained which demonstrate the so-called scin-

tillation effect which is a term used to describe the fact that the azimuth

estimate lies outside the azimuth interval ( Bs, f31).

16

-IEzm

SIR (C

ATCRBS AZIMUTH

1

ADABS AZIMUTH

SNR = 20dB0,=0

81 = 0,5Bw

I

0 Tl Z7T

-m

-40

-3

3

to

.+m

RELATIVE PHASE BETwEEN TARGET AND INTERFERENCE SIGNALS (rod)

Fig. 2. Azimuth estimate in the presence of

17

interference.

4

v,—3E

:m

Lomu0

2

0

-2

.,

lEzEuIl,.NO FADING

FAOING

h /

I \

I I

SIR= -fdB

/:~:::~H

\ DABS AZIMUTHSNR= 20d B

0,=0.5BwSIR =ld B

\1“/

&. .-“ ,, c ,,

RELATIVE Pti ASF BETWEEN TARGET AND INTERFERENCE SIG NALS (rod)

Fig. 3. Azimuth estimate in the presence of interference.

18

(d) Case 4: Effects of Relative Phase

It is reasonable to model the phase as a uniformly distributed

random variable on (O, 2TT). Then averaging over phase, it is easy to show

,. that

J_

2TI

‘h

b(v) d~ =

o ifpo<l

1 ifp>l0

(35)

In a practical situation, the key issue is how correlated the phase is from

sample to sample within a reply. For multipath, the correlation time can

be several seconds because the phase relationship depends on the relative

path lengths between the direct and multipath signals and for typical aircraft

speeds, these change slowly. For ATCRBS interference, the phase difference

will be independent from sample to sample since lhe transmitter tube is in-

coherent from pulse to pulse. IIowcver, since there arc relatively few inter-

ference samples in any one reply, the inherent averaging due to phase

cannot reliably be exploited and wc must face the possibility of having to

deal withthe large errors shown in Figure 2.

19

(e) Case 5: Sidelobe Intcrfcrcnce

Although all of the results given so far are applicable to mainbcam

and sidelobe inter fcrcncc, there arc some remarks that are worth noting for

the latter case. In the case of A’1’CR}~S sidelobe inter fercncc, it is possible

that A1/A~ can be much larger than unity. I,ct us SUpPOS~ fOr example that the

DABS target is at maximum range (100 mi) and the A1’CRl\S interferer is at

the same power, but at a C.1OSC in range (10 mi say). Then the 20 dl’J decrease

in ISR due to the antenna sidelobes is compensated by the 20 cl]> increase in

power due to the range diffcrcnc. c. I’hcn Do can be near O dR and since the

equivalent azimuth is of the orclcr of 1 or 2 beamwidths then, as we have

seen, large bias errors can be cxpcctcct. Therefore wc conclude that if a

strong AT CRBS signal is being reccivcd in a near-in siclclobc, the monopulsc

azimuth estimate can have a lar~c bias that exhibits the same clependcncc on

the relative phase bctwccn the DABS and AT’CR13S transponders that was

shownto exist for mainbcam interference. Furthermore, it is clear that very

low sidelobcs is not a sufficient means of eliminating the effects of this inter-

ference. Therefore, at least one additional lCVC1 of data processing will be

needed to improve the quality of the azimuth estimate for tbcse cases. T’his

is referred to as monopulsc data editing and will be discussed in a later

paper [4].

20

I .

For sidelobe multipath, on the other hancl, wc can reasonably expect

that the reflection coefficient, A1/A - .707. For antenna patterns with 20 dBs

peak sidelobe levels, this puts PO less than . 07. From our studies so far we

know that the bias error peaks when the direct and multipath signals are w out

of phase. Using (34) the bias error can be bounded by

(36)

where the last approximation follows from the fact that o is quite a bit lCSSo

than unity, For a DABS target on horcsight, 86 = O and then using (23) and (27)

the bound be comes

(37)

which shows that the bias error depends on the sidclobe level Of the differ-

ence pattern relative to the sum beam gain, For the example studied, k = 1. 5,

A1/A~ .. 707 and G2( 91)/ Gl(~~) = . 1, which yields a peak bias error .047

3 dB beamwidt.hs.

4, INTERFERENCE EFFECTS ON ESTIMATE VARIANCE

In this section we will compute the variance in the monopulse azimuth

estimate. Sharenson [3] has studied this problem for the case when the inter-

ference consisted only of receiver noise. We shall extend his analysis to in-

clude the effects of mainbeam and sidelobe interference. We begin with tbe

equivalent signal model formulated in (l), (2), and (5). The monopulse estim-

ator of interest is

21

(38)

which gives the azimuth estimate in 3 dl> beamwidths. l“rom (1) and (6),

so that

yi = lJi + n.1

lY1f= lu~

where the last approximation holds provided t.hc equivalent SNR, IU1 [ 2/202

is large enough, typically >12 dB. Then the moments of (38) can be com-

puted by evaluating the moments of

>k

7, = Rc(ylyz ) (41 )

Since the noise terms nl and nz are inclcpcndcmt, the first moment is simply

.,.

( ‘)z ‘ “ul~z(42)

and hence the first moment of the azimuth estimate is given approximately by

K ‘2e=&c0s(B2-P1)

1

(43)

22

. .

which is the same result wc obtained in the last section when the effective

SNR was greater than 8 dB. Proceeding further, the second moment of (41)

>~can be calculated using the identity Re(x)R@ = $ Re(x Y ) + ~Re(x Y.) In this

>:case we let x= y. ylyz and then

— ——

2z= ~ly1121y2]2+~Rey~y~2

2 TFrom (39) and using the fact that Inil = 202, ni = O, wc obtain

lYi12 = lui12 + 204

72Yi ,= LJ.

Combining (42), (44) and (45 ) wc can show that the variance of z is

~ -2-z

(= IIJ

(44 )

(45a)

(45b)

)12+. \u212 02+204 (46)

Finally we use (4o), (M ) and (M ) tO shOw that the variance Of the azimuth

estimate is

,-[14.-+.-]Var (6) = ‘z (47)

23

Since [ui I = Vi, then (47) could bc expressed in terms of the complex antenna

pattern parameters by using (14). Unfortunately the resulting expressions

are complicated and in order to obtain sornc ~meaningful analytical statements

it is not fruitful to retain the most general equations. Therefore, we shall

.,follow the direction of the proceeding section and restrict our attention to the

case of real antenna patterns. In addition wc note that (47) describes the‘,

estimate variance only when the cffectivc SNR is large, This means that

lu112/202>x, hence wc can also neglect the effects of the second order

SNR term that appears in (47). Under these conditions we can then use (14)

for real antenna patterns and show that the variance reduces to

1 +k2’Ff~i$l~- 2

z l+k61

22l+2p cos~ig

2 l+k 9 0 l+k2’r3j0

1 + k2Ei2Var (~). z: .

s s(48)

A~G1(9~) k22

l+2poc0sq+po

where, as in the case of the bias effects, O. ‘“ arc given by (23) and (27).and 01

In the next few paragraphs we shall consider some more specialized

,,cases.

(a) Case 1: No Intcrfcrencc.,

When there is no intcrfercncc, AI . 0, hcncc p . 0 and from (48)0

we sce that the variance is simply

24

2 1 + k28;a

Var ($) = .

A~G;(8~) k2(49)

which is the result obtained by Sharenson [3] and others, and shows that the

variance decreases with SNR and the slope of the normalized difference

pattern and increases with the degree to which the target is off bores ight.. .

(b) Case 2: Fading

If the target and interference arrive from the same azimuth

(multipath from a flat earth for example) then %1 = 91 = eq and (48) becomes

2 l+k282

Var ($) =a s 1

(50)

A~G; (96) k2 1 + 2 PO COSO+. P20

which shows that when multipath fades occur there can be a reduction in the

effective SNR which in turn leads to an incrcasc in the variance. In the last

section we found that a bias error was also introduced in the fading situation.

(c) Case 3: Azimuthal Variance

When the target and interferer are at different azimuths we see

from (46) that the variance will bc further increased. Typical results arc

.,plotted in Figure 4 for the case of a target on boresight at 20 dB SNR

and an interferer located at the edge of the 3 dB beamwidth. We see that the

25

0EOa0

SNR=20d B

Q*=O

t?l =0.5Bw fl SIR (dB)

(

Za 0 0.577 n 1.5T 2T~

RELATIvE PHASE BETWEEN TARGET ANO INTERFERENCE SIG NALS(rad)

Fig. 4. Variance of the azimuth estimate in the presence of interference.

most significant degradation in performance occurs at the out-of-phase con-

dition. However, the magnitude of the errors is much smaller than the con-

tribution due to the bias, and furthermore since the errors are random,

their effect can bc further reduced simply by averaging many

,. Therefore, we conclude that the bias error is likely to be the

,. problem for ATC direction finding.

of the estimates.

more troublesome

5. CONCLUSIONS

An exact expression for the bias of a two beam monopulse azimuth

estimate has been derived that describes the degradation in performance due

to receiver noise and AT’ CRBS and multipath intcrfcrencc. At low SNR and

high ISR the bias tends toward borcsight, although for all practical purposes

the net effect of this bias is negligible. More important is the azimuthal bias

that describes tbe so-called scintillation of the azimuth estimate that has

been observed in many monopulsc tracking problems. Depending on the

values of the SNR, ISR and the relative phase between the DA}IS and inter-

ference signals, the bias can bc quite significant even when the inter fcri.ng

signal arrives through a low lCVC1 sidclobe,

A first order analysis was used to obtain the variance in the.

azimuth estimate when intcrfcrcmcc is present in a noisy background.

Although the results indicate that the random errors will not be insignifi-

cant, the deterministic. bias error will bc, by far, the more dominant effect.

For ISR’S less than -30 cl}~, the effect of the interference is negligible.

As this quantity increases, the bias and variance increase and errors that are

of the order of a beamwidth can bc obtained. For ISR’ s between + 10 dIl there—

is a strong dependence on the relative phase, with a peak error occurring at

the out-of-phase condition. As the ISR is further inc. rcased, this peaking

effect subsides until the intcrfcrcncc completely captures the monopulse>,

processor which at this point WOUIC1track the intcrfcrcncc target..,

Although the results arc applicable to analyzing the effects of

A1’CRBS interference and multipath, a distinction between the two phenomena

should be noted. For mu]tipath, the direct ancl indirect signals are at the

same frequency and coherent in the sense that their relative phase may be

constant during several seconds duration. For A’1’CRIH interference,

however, the transponders arc incoherent from bit to bit and possibly for

samples within a bit since there may bc carrier frequency offset. ‘1’hercforc,

in the AT CRBS case, there will bc averaging that can be exploited to reduce

the overall bias error, in which case the effect of the increased error

variance would become a more important effect. From a processing point

of view, this could be ovcrcomc by using more samples to form the azimuth

estimate.

In a separate study [4] the pcrformanc. c of the maximum likelihood .

interference detector has been prescmtcd in detail. T’he next step is to com -

bine the results of both studies to evaluate the data editing concept. T’his

will determine whether there is any promise to the idea of introducing an

additional level of data processing to improve the overall quality of the

azimuth c stimatc.

28

1. 1>. 11. Pruslinr l,M”ltipath.Ikllse Monopulse Accuracy, “

Lincoln l.,aboratory “1’ethnical Report 487, (I9 November1971). DDC AI>-737369.

2. A.A. L. l\rownc, l,incoln J,aboratory Internal, Memorandum.

3. S. Sharenson, “Angle Wtimination Accuracy with a MonopulseRadar inthc Search Mode, ,HIRE Transactions on Aerospace

and Navigational Electronics (September 1962), pp. 175-179.

4. R. J. McAulay and 1’. P. McGarty, “Maximum LikelihoodDetection of Unresolved Radar Targets ancl Multipath, “ tobe published.

5. IJ, K. Barton, “Raclar System Analysis, “I’rentice - Ilall, NewJersey, 1964.

. .

APPENDIX

To compute the average of (9) with respect to al, a uniformly

distributed random variable on (O, 2TT)we have

2Tl 2

E (k~) = C0S(~2- ~1)

.r

‘l+vl NlcOswl

% ‘2d al

20 V;+N1+2V1N1 .Os O1 I

Since

zero.

the integrand of the second term is an odd function, its integral is

Using standard integral tables, 13rowne showed that the integral in the

first term reduces to

V; -N:2TI

‘2.os(~-B1) -&. ~

f

1da

( )1

V;+N; o ~+2V1N1 1

— Cosci

V;+N:1

1 ‘2

~[

V; -N;—.l-— COS(B2-B1) 1+ V2+N2

7 VI

11

30

~v2N2

II1/2

11

(V:+ N;)2

.

where the positive square root is under stood.. Therefore

if N1<V1

as given in. (10).

ACKNOWLEDGMENT

The author would like to acknowledge the unpublished work of A. A. 1..

13rowne of the Lincoln Laboratory who first derived the exact expression for

the first moment of the monopulse estimator in a receiver noise background.

The author would also like to thank D. F. DeLong who provided Brownc’s

obscure but important memo and whose comments, along with those of

E. Kelly, resulted in a considerable improvement in the final draft. Final~y.

the author acknowledges the related but unpublished work done in this area

by J. Modestino.

31